The present invention relates to RF transmission systems and methods, and more particularly, to apparatus, algorithms, techniques and methods that implement constrained clipping for peak-to-average power ratio (or crest factor) reduction in multicarrier (MC) systems. Multicarrier signals are a popular choice in modern communications systems. Several specific multicarrier communications signals include orthogonal frequency division multiplexing (OFDM), orthogonal frequency division multiple access (OFDMA), multicarrier code division multiple access (MC-CDMA) and discrete multi-tone (DMT).
Despite their popularity, multicarrier signals have the drawback of large peak-to-average power ratios (PARs) or equivalently, crest factors. There have been a number of proposals to solve this problem, many of which require receiver-side modifications that may be difficult to deploy in existing communications systems. Such approaches include selected mapping, discussed by R. W. Bauml, R. F. H. Fischer and J. B. Huber, “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,” IEE Electronics Letters, vol. 32, no. 22, pp. 2056-2057, October 1996, and R. J. Baxley and G. T. Zhou, “MAP metric for blind phase sequence detection in selected mapping,” IEEE Trans. on Broadcasting, vol. 51, no. 4, pp. 565-570, December 2005, companding discussed by X. Wang, T. Tjhung, and C. Ng, “Reduction of peak-to-average power ratio of OFDM system using a companding technique,” IEEE Trans. on Broadcasting, vol. 45, pp. 303-307, September 1999, partial transmit sequence discussed by S. Muller and J. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” IEE Electronics Letters, vol. 33, pp. 368-369, February 1997, tone injection, tone reservation, or coding as discussed in J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSL and Wireless, Kluwer Academic Publishers, 2000 and references therein. Additionally, there have been several methods proposed that do not require receiver-side modification such as clipping-based methods, active constellation extension (ACE) discussed by B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation extension,” IEEE Trans. on Broadcasting, vol. 49, no. 3, pp. 258-268, September 2003, and an ACE-like method proposed by S.-K. Deng and M.-C. Lin, “OFDM PAPR reduction using clipping with distortion control,” Proc. IEEE Intl. Conf. Communications, pp. 2563-2567, May 2005.
Orthogonal frequency division multiplexing (OFDM) is a particularly popular multicarrier transmission method in high-speed communications schemes. For example, the IEEE 802.11 wireless standard has an OFDM option and WiMax (IEEE 802.16) also uses OFDM or OFDMA. The IEEE 802.16 wireless standard is described in “IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems,” IEEE Std. 802.16-2004 (Revision of IEEE Std. 802.16-2001), pp. 1-857, 2004. Additionally, OFDM has been adopted by digital audio broadcasting (DAB), digital video broadcasting (DVB) and high performance radio metropolitan area network (HIPERMAN). To illustrate the current algorithm, its implementation is demonstrated in an OFDM system. However, the algorithm is readily adaptable to any multicarrier communications system.
Overview of Clipping Techniques
Let {Xk}k=−N/2N/2-1 be the frequency domain sequence of an OFDM symbol where N is the number of subcarriers. Since Nyquist rate samples might not represent the peaks of the continuous-time signal, it is desirable to show CFR performance on over-sampled discrete-time signals. This is discussed by J. Armstrong, “New OFDM peak-to-average power reduction scheme,” Proc. IEEE VTS 53rd Vehicular Technology Conference. vol. 1, pp. 756-760, May 2001, J. Armstrong, “Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering,” IEE Electronics Letters, vol. 38, issue 5, pp. 246-247, February 2002, X. Li and L. J. Cimini, “Effects of clipping and filtering on the performance of OFDM,” Proc. VTC'97, pp. 1634-1638, May 1997, H. L. Maattanen, N. Y. Ermolova and S. G. Haggman, “Nonlinear amplification of clipped-filtered multicarrier signals,” IEEE VTS 61st Vehicular Technology Conference, vol. 2, pp. 958-962, May 2005, for example. It is typical to use an over-sampling factor of L≧4 so that the PAR before the digital to analog (D/A) conversion can accurately describe the PAR after the D/A conversion, such as is discussed by M. Sharif and B. H. Khalaj, “Peak to mean envelope power ratio of over-sampled OFDM signals: an analytical approach,” IEEE International Conference on Communications, vol. 5, pp. 1476-1480, June 2001, for example. For CFR methods with distortion, over-sampling is also necessary to examine the out-of-band spectral characteristics of the signal after CFR.
Define the out-of-band indices to be the set O: [−LN/2,−N/2−1]∪[N/2,LN/2−1] and the in-band indices to be the set I: [−N/2,N/2−1]. Denote the zero-padded version of
                                          X            k                    ⁢                                          ⁢          by          ⁢                                          ⁢                                    {                              X                k                                  (                  L                  )                                            }                                      k              =                                                -                  L                                ⁢                                                                  ⁢                                  N                  /                  2                                                                                    L                ⁢                                                                  ⁢                                  N                  /                  2                                            -              1                                ⁢                                          ⁢          where                ⁢                                  ⁢                              X            k                          (              L              )                                =                      {                                                                                                      X                      k                                        ,                                                                                                              k                      ∈                      I                                        ,                                                                                                                    0                    ,                                                                                        k                    ∈                                          O                      .                                                                                                                              (        1        )            
The over-sampled discrete-time symbol xn(L) can be calculated as follows:
                                          x            n                          (              L              )                                =                                    1                                                L                  ⁢                                                                          ⁢                  N                                                      ⁢                                          ∑                                  k                  =                                                            -                      L                                        ⁢                                                                                  ⁢                                          N                      /                      2                                                                                                            L                    ⁢                                                                                  ⁢                                          N                      /                      2                                                        -                  1                                            ⁢                                                X                  k                                      (                    L                    )                                                  ⁢                                  ⅇ                                      j                    ⁢                                                                  2                        ⁢                        π                        ⁢                                                                                                  ⁢                        k                        ⁢                                                                                                  ⁢                        n                                                                    L                        ⁢                                                                                                  ⁢                        N                                                                                                                                ,                  0          ≤          n          ≤                                    L              ⁢                                                          ⁢              N                        -            1.                                              (        2        )            
Clipping is the simplest CFR method. Polar clipping xn(L) with threshold Amax yields
                                          x            _                    n                      (            L            )                          =                  {                                                                                          x                    n                                          (                      L                      )                                                        ,                                                                                                                                            x                      n                                              (                        L                        )                                                                                                  ≤                                      A                                          max                      ⁢                                                                                          ,                                                                                                                                                                                      A                      max                                        ⁢                                          ⅇ                                              j∠                        ⁢                                                                                                  ⁢                                                  x                          n                                                      (                            L                            )                                                                                                                                ,                                                                                                                                            x                      n                                              (                        L                        )                                                                                                  >                                                            A                      max                                        .                                                                                                          (        3        )            
The corresponding frequency domain signal is
                                                                        X                _                            k                              (                L                )                                      =                                          1                                                      L                    ⁢                                                                                  ⁢                    N                                                              ⁢                                                ∑                                      n                    =                    0                                                                              L                      ⁢                                                                                          ⁢                      N                                        -                    1                                                  ⁢                                                                            x                      _                                        n                                          (                      L                      )                                                        ⁢                                      ⅇ                                                                  -                        j                                            ⁣                                                                        2                          ⁢                          π                          ⁢                                                                                                          ⁢                          k                          ⁢                                                                                                          ⁢                          n                                                                          L                          ⁢                                                                                                          ⁢                          N                                                                                                                                                  ,          -                ⁣                                            L              ⁢                                                          ⁢              N                        2                    ≤          k          ≤                                                    L                ⁢                                                                  ⁢                N                            2                        -            1.                                              (        4        )            
The clipping operation in equation (3) generates distortions in Xk(L) both in-band and out-of-band. In-band distortion is observed when Xk(L)≠Xk for k∈I. Out-of-band spectral regrowth is revealed since Xk(L)≠0 for k∈O. These are in contrast to the unclipped signal Xk(L) described in equation (1).
Denote by
                                                                                          E                  k                                =                                ⁢                                                                            X                      _                                        k                                          (                      L                      )                                                        -                                      X                    k                                          (                      L                      )                                                                                  ,                                                                        ⁢                              k                ∈                I                                                                                                        =                                ⁢                                                                            X                      _                                        k                                          (                      L                      )                                                        -                                      X                    k                                                              ,                                                                        ⁢                              k                ∈                I                                                                        (        5        )            the error vector at the kth subcarrier in-band. The formula for calculating the so-called error vector magnitude (EVM) varies depending on the communication standard (see “IEEE Standard for Local and Metropolitan Area Networks Part 16 . . . ”, “Radio transmission and reception,” GSM Recommendation 05.05, December 1999, and “Base station (BS) conformance testing (FDD),” 3GPP TS 25.141, v 3.14.0, pp. 1-109, May 2005, for instance). As an example, using the EVM metric defined in the WiMax standard,
                                          EVM            ⁢                          {                                                x                  _                                n                                  (                  L                  )                                            }                                =                                    1                              S                max                                      ⁢                                                            1                  N                                ⁢                                                      ∑                                          k                      ∈                      I                                                        ⁢                                                                                                          E                        k                                                                                    2                                                                                      ,                            (        6        )            where Smax is the maximum amplitude of the constellation (see “IEEE Standard for Local and Metropolitan Area Networks Part 16 . . . ”). In other words, EVM is a scaled root-mean-squared (rms) distance between the desired constellation points Xk and positions of the signal Xk(L), k∈I.
The EVM calculated according to equation (6) is only for one symbol period. However, in some OFDM standards the measured period may contain several OFDM symbols and the EVM is taken as the average. When N is large, the per symbol EVM will be very close to the average EVM over several symbols according to the law of large numbers.
Recall that xn(L) is the result of simple clipping. Suppose that {tilde over (x)}n(L) is the signal that actually gets transmitted, which may be obtained after certain operations on xn(L). The standard usually specifies a threshold Th for the EVM and a spectral mask P(ω) for the power spectral density (PSD) of the transmitted signal {tilde over (x)}n(L). It would be desirable to obtain {tilde over (x)}n(L), or equivalently, {tilde over (X)}k(L), such that:
(i) PAR{{tilde over (x)}n(L)}<<PAR{xn(L)};
(ii) EVM{{tilde over (x)}n(L)}≦Th;
(iii) PSD{{tilde over (x)}n(L)}≦P(ω), for
      π    L    <          ω        <      π    .  
In calculating EVM{{tilde over (x)}n(L)}, replace the Ek in equation (6) by{tilde over (E)}k={tilde over (X)}k(L)−Xk,k∈I.  (7)
One well-known method to contain the out-of-band spectral regrowth (i.e., objective (iii)) is to set {tilde over (X)}k(L)=0,∀k∈O; this is the so-called frequency domain filtering method proposed by J. Armstrong in “New OFDM peak-to-average power reduction scheme,” Proc. IEEE VTS 53rd Vehicular Technology Conference. vol. 1, pp. 756-760, May 2001. With the Armstrong method, nothing is done to control the in-band EVM (i.e., objective (ii)). The out-of-band spectral regrowth stays far below the spectrum mask (0<<P(ω)), which essentially wastes energy that is allotted by the standard that could be used for CFR (i.e., objective (i)). After filtering, the PAR is always larger than that of the simple clipping method; i.e., PAR{{tilde over (x)}n(L)}>PAR{{tilde over (x)}n(L)}.
It would be desirable to improve upon Armstrong's work by incorporating the EVM constraint and by being more efficient with out-of-band energy allocation.